Multigrid computational methods are

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1 M ULTIGRID C OMPUTING Wy Multigrid Metods Are So Efficient Originally introduced as a way to numerically solve elliptic boundary-value problems, multigrid metods, and teir various multiscale descendants, ave since been developed and applied to various problems in many disciplines. Tis introductory article provides te basic concepts and metods of analysis and outlines some of te difficulties of developing efficient multigrid algoritms /06/$ IEEE Copublised by te IEEE CS and te AIP IRAD YAVNEH Multigrid computational metods are well known for being te fastest numerical metods for solving elliptic boundary-value problems. Over te past 30 years, multigrid metods ave earned a reputation as an efficient and versatile approac for oter types of computational problems as well, including oter types of partial differential equations and systems and some integral equations. In addition, researcers ave successfully developed generalizations of te ideas underlying multigrid metods for problems in various disciplines. (See te Multigrid Metods Resources sidebar for more details.) Tis introductory article presents te fundamentals of multigrid metods, including explicit algoritms, and points out some of te main pitfalls using elementary model problems. Tis material is mostly intended for readers wo ave a practical interest in computational metods but little or no acquaintance wit multigrid tecniques. Te article also provides some background for tis special issue and oter, more advanced publications. Tecnion Israel Institute of Tecnology Basic Concepts Let s begin wit a simple example based on a problem studied in a different context. Global versus Local Processes Te ometown team as won te regional cup. Te players are to receive medals, so it s up to te coac to line tem up, equally spaced, along te goal line. Alas, te field is muddy. Te coac, wo abors mud, tries to accomplis te task from afar. He begins by numbering te players 0 to N and orders players 0 and N to stand by te left and rigt goal posts, respectively, wic are a distance L apart. Now, te coac could, for example, order player i, i =,, N, to move to te point on te goal line tat is at a distance il/n from te left goal post. Tis would be a global process tat would solve te problem directly. However, it would require eac player to recognize te left-and goal post, perform some fancy aritmetic, and gauge long distances accurately. Suspecting tat is players aren t up to te task, te coac reasons tat if eac player were to stand at te center point between is two neigbors (wit players 0 and N fixed at te goal posts), is task would be accomplised. From te local rule, a global order will emerge, e pilosopizes. Wit tis in mind, te coac devises te following simple iterative local process. At eac iteration, e blows is wistle, and player i, i =,, N, COMPUTING IN SCIENCE & ENGINEERING

3 (a) (b) (c) Figure. Multiscale player alignment. Red disks sow te current position, and blue disks sow te previous position, before te last wistle blow: (a) large, (b) medium, and (c) small scales. vergent tat is, te errors in te players positions will eventually be as small as we wis to make tem. However, tis migt take a long time. Suppose te players positions are as sown in Figure c. In tis case, te players move sort distances at eac iteration. Sadly, tis slow crawl toward convergence will persist for te duration of te process. In contrast, convergence is obtained in a single iteration in te position sown in Figure d. Wat is te essential property tat makes te convergence slow in te position of Figure c, yet fast in Figure d? In Figure c, we ave a largescale error. But Jacobi s process is local, employing only small-scale information tat is, eac player s new position is determined by is near neigborood. Tus, to eac player, it seems (from is myopic viewpoint) tat te error in position is small, wen in fact it isn t. Te point is tat we cannot correct wat we cannot detect, and a large-scale error can t be detected locally. In contrast, Figure d sows a small-scale error position, wic can be effectively detected and corrected by using te local process. Finally, consider te position in Figure e. It clearly as a small-scale error, but Jacobi s process doesn t reduce it effectively; rater, it introduces an oversoot. Not every local process is suitable. However, tis problem is easily overcome witout compromising te process s local nature by introducing a constant damping factor. Tat is, eac player moves only a fraction of te way toward is designated target at eac iteration; in Figure f, for example, te players movements are damped by a factor /3. Multiscale Concept Te main idea beind multigrid metods is to use te simple local process but to do so at all scales of te problem. For convenience, assume N to be a power of two. In our simple example, te coac begins wit a large-scale view in wic te problem consists only of te players numbered 0, N/, and N. A wistle blow puts player N/ alfway between is large-scale neigbors (see Figure a). Next, addressing te intermediate scale, players N/4 and 3N/4 are activated, and tey move at te sound of te wistle to te midpoint between teir mid-scale neigbors (see Figure b). Finally, te small scale is solved by Jacobi s iteration at te remaining locations (Figure c). Tus, in just tree wistle blows generally, log (N) and only N individual moves, te problem is solved, even toug te simple local process as been used. Because te players can move simultaneously at eac iteration in parallel, so to speak te number of wistle blows is important in determining te overall time, and log (N) is encouraging. For million players, for example, Jacobi s process would require a mere 0 wistle blows. Researcers ave successfully exploited te idea of applying a local process at different scales in many fields and applications. Our simple scenario displays te concept but ides te difficulties. In general, developing a multiscale solver for a given problem involves four main tasks: coosing an appropriate local process, coosing appropriate coarse (large-scale) variables, coosing appropriate metods for transferring information across scales, and developing appropriate equations (or processes) for te coarse variables. Depending on te application, eac of tese tasks can be simple, moderately difficult, or extremely callenging. Tis is partly te reason wy multigrid continues to be an active and triving researc field. We can examine tese four tasks for developing a multiscale solver by using a sligtly less trivial problem. Before doing tis, let s analyze Jacobi s player-alignment algoritm and give some matematical substance to our intuitive notions on te dependence of its efficiency on te scale of te error. 4 COMPUTING IN SCIENCE & ENGINEERING

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